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Theorem ushgredgedgloop 26314
Description: In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex and the set of loops at this vertex. (Contributed by AV, 11-Dec-2020.)
Hypotheses
Ref Expression
ushgredgedgloop.e 𝐸 = (Edg‘𝐺)
ushgredgedgloop.i 𝐼 = (iEdg‘𝐺)
ushgredgedgloop.v 𝑉 = (Vtx‘𝐺)
ushgredgedgloop.a 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}
ushgredgedgloop.b 𝐵 = {𝑒𝐸𝑒 = {𝑁}}
ushgredgedgloop.f 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
Assertion
Ref Expression
ushgredgedgloop ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐵,𝑒   𝑒,𝐸,𝑖   𝑒,𝐺,𝑖,𝑥   𝑒,𝐼,𝑖,𝑥   𝑒,𝑁,𝑖,𝑥   𝑒,𝑉,𝑖,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑒,𝑖)   𝐵(𝑥,𝑖)   𝐸(𝑥)   𝐹(𝑥,𝑒,𝑖)

Proof of Theorem ushgredgedgloop
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2752 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
2 ushgredgedgloop.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2ushgrf 26149 . . . 4 (𝐺 ∈ USHGraph → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
43adantr 472 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
5 ssrab2 3820 . . 3 {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ⊆ dom 𝐼
6 f1ores 6304 . . 3 ((𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ⊆ dom 𝐼) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}))
74, 5, 6sylancl 697 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}))
8 ushgredgedgloop.f . . . . 5 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
9 ushgredgedgloop.a . . . . . . 7 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}
109a1i 11 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}})
11 eqidd 2753 . . . . . 6 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ 𝑥𝐴) → (𝐼𝑥) = (𝐼𝑥))
1210, 11mpteq12dva 4876 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑥𝐴 ↦ (𝐼𝑥)) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ↦ (𝐼𝑥)))
138, 12syl5eq 2798 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ↦ (𝐼𝑥)))
14 f1f 6254 . . . . . . 7 (𝐼:dom 𝐼1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
153, 14syl 17 . . . . . 6 (𝐺 ∈ USHGraph → 𝐼:dom 𝐼⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
165a1i 11 . . . . . 6 (𝐺 ∈ USHGraph → {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ⊆ dom 𝐼)
1715, 16feqresmpt 6404 . . . . 5 (𝐺 ∈ USHGraph → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ↦ (𝐼𝑥)))
1817adantr 472 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}) = (𝑥 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ↦ (𝐼𝑥)))
1913, 18eqtr4d 2789 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹 = (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}))
20 ushgruhgr 26155 . . . . . . . 8 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
21 eqid 2752 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
2221uhgrfun 26152 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2320, 22syl 17 . . . . . . 7 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
242funeqi 6062 . . . . . . 7 (Fun 𝐼 ↔ Fun (iEdg‘𝐺))
2523, 24sylibr 224 . . . . . 6 (𝐺 ∈ USHGraph → Fun 𝐼)
2625adantr 472 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → Fun 𝐼)
27 dfimafn 6399 . . . . 5 ((Fun 𝐼 ∧ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ⊆ dom 𝐼) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑒})
2826, 5, 27sylancl 697 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}) = {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑒})
29 fveq2 6344 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
3029eqeq1d 2754 . . . . . . . . . 10 (𝑖 = 𝑗 → ((𝐼𝑖) = {𝑁} ↔ (𝐼𝑗) = {𝑁}))
3130elrab 3496 . . . . . . . . 9 (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}))
32 simpl 474 . . . . . . . . . . . . . . 15 ((𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) → 𝑗 ∈ dom 𝐼)
33 fvelrn 6507 . . . . . . . . . . . . . . . 16 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran 𝐼)
342eqcomi 2761 . . . . . . . . . . . . . . . . 17 (iEdg‘𝐺) = 𝐼
3534rneqi 5499 . . . . . . . . . . . . . . . 16 ran (iEdg‘𝐺) = ran 𝐼
3633, 35syl6eleqr 2842 . . . . . . . . . . . . . . 15 ((Fun 𝐼𝑗 ∈ dom 𝐼) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
3726, 32, 36syl2an 495 . . . . . . . . . . . . . 14 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁})) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
38373adant3 1126 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → (𝐼𝑗) ∈ ran (iEdg‘𝐺))
39 eleq1 2819 . . . . . . . . . . . . . . 15 (𝑓 = (𝐼𝑗) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4039eqcoms 2760 . . . . . . . . . . . . . 14 ((𝐼𝑗) = 𝑓 → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
41403ad2ant3 1129 . . . . . . . . . . . . 13 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ (𝐼𝑗) ∈ ran (iEdg‘𝐺)))
4238, 41mpbird 247 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → 𝑓 ∈ ran (iEdg‘𝐺))
43 ushgredgedgloop.e . . . . . . . . . . . . . . . 16 𝐸 = (Edg‘𝐺)
44 edgval 26132 . . . . . . . . . . . . . . . . 17 (Edg‘𝐺) = ran (iEdg‘𝐺)
4544a1i 11 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4643, 45syl5eq 2798 . . . . . . . . . . . . . . 15 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
4746eleq2d 2817 . . . . . . . . . . . . . 14 (𝐺 ∈ USHGraph → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
4847adantr 472 . . . . . . . . . . . . 13 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
49483ad2ant1 1127 . . . . . . . . . . . 12 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑓 ∈ ran (iEdg‘𝐺)))
5042, 49mpbird 247 . . . . . . . . . . 11 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → 𝑓𝐸)
51 eqeq1 2756 . . . . . . . . . . . . . . 15 ((𝐼𝑗) = 𝑓 → ((𝐼𝑗) = {𝑁} ↔ 𝑓 = {𝑁}))
5251biimpcd 239 . . . . . . . . . . . . . 14 ((𝐼𝑗) = {𝑁} → ((𝐼𝑗) = 𝑓𝑓 = {𝑁}))
5352adantl 473 . . . . . . . . . . . . 13 ((𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) → ((𝐼𝑗) = 𝑓𝑓 = {𝑁}))
5453a1i 11 . . . . . . . . . . . 12 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) → ((𝐼𝑗) = 𝑓𝑓 = {𝑁})))
55543imp 1101 . . . . . . . . . . 11 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → 𝑓 = {𝑁})
5650, 55jca 555 . . . . . . . . . 10 (((𝐺 ∈ USHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) ∧ (𝐼𝑗) = 𝑓) → (𝑓𝐸𝑓 = {𝑁}))
57563exp 1112 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}) → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑓 = {𝑁}))))
5831, 57syl5bi 232 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} → ((𝐼𝑗) = 𝑓 → (𝑓𝐸𝑓 = {𝑁}))))
5958rexlimdv 3160 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓 → (𝑓𝐸𝑓 = {𝑁})))
60 funfn 6071 . . . . . . . . . . . . . 14 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
6160biimpi 206 . . . . . . . . . . . . 13 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
6223, 61syl 17 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
63 fvelrnb 6397 . . . . . . . . . . . 12 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6462, 63syl 17 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓))
6534dmeqi 5472 . . . . . . . . . . . . . . . . . . . . . 22 dom (iEdg‘𝐺) = dom 𝐼
6665eleq2i 2823 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ dom (iEdg‘𝐺) ↔ 𝑗 ∈ dom 𝐼)
6766biimpi 206 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ dom (iEdg‘𝐺) → 𝑗 ∈ dom 𝐼)
6867adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → 𝑗 ∈ dom 𝐼)
6968adantl 473 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ dom 𝐼)
7034fveq1i 6345 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((iEdg‘𝐺)‘𝑗) = (𝐼𝑗)
7170eqeq2i 2764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = ((iEdg‘𝐺)‘𝑗) ↔ 𝑓 = (𝐼𝑗))
7271biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = ((iEdg‘𝐺)‘𝑗) → 𝑓 = (𝐼𝑗))
7372eqcoms 2760 . . . . . . . . . . . . . . . . . . . . . . 23 (((iEdg‘𝐺)‘𝑗) = 𝑓𝑓 = (𝐼𝑗))
7473eqeq1d 2754 . . . . . . . . . . . . . . . . . . . . . 22 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} ↔ (𝐼𝑗) = {𝑁}))
7574biimpcd 239 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = {𝑁} → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = {𝑁}))
7675adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = {𝑁}))
7776adantld 484 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼𝑗) = {𝑁}))
7877imp 444 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼𝑗) = {𝑁})
7969, 78jca 555 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ dom 𝐼 ∧ (𝐼𝑗) = {𝑁}))
8079, 31sylibr 224 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → 𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}})
8170eqeq1i 2757 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐺)‘𝑗) = 𝑓 ↔ (𝐼𝑗) = 𝑓)
8281biimpi 206 . . . . . . . . . . . . . . . . . 18 (((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝐼𝑗) = 𝑓)
8382adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝐼𝑗) = 𝑓)
8483adantl 473 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝐼𝑗) = 𝑓)
8580, 84jca 555 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓)) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ∧ (𝐼𝑗) = 𝑓))
8685ex 449 . . . . . . . . . . . . . 14 ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = 𝑓) → (𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} ∧ (𝐼𝑗) = 𝑓)))
8786reximdv2 3144 . . . . . . . . . . . . 13 ((𝐺 ∈ USHGraph ∧ 𝑓 = {𝑁}) → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓))
8887ex 449 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (𝑓 = {𝑁} → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓)))
8988com23 86 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = 𝑓 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓)))
9064, 89sylbid 230 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (𝑓 ∈ ran (iEdg‘𝐺) → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓)))
9147, 90sylbid 230 . . . . . . . . 9 (𝐺 ∈ USHGraph → (𝑓𝐸 → (𝑓 = {𝑁} → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓)))
9291impd 446 . . . . . . . 8 (𝐺 ∈ USHGraph → ((𝑓𝐸𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓))
9392adantr 472 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → ((𝑓𝐸𝑓 = {𝑁}) → ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓))
9459, 93impbid 202 . . . . . 6 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓 ↔ (𝑓𝐸𝑓 = {𝑁})))
95 vex 3335 . . . . . . 7 𝑓 ∈ V
96 eqeq2 2763 . . . . . . . 8 (𝑒 = 𝑓 → ((𝐼𝑗) = 𝑒 ↔ (𝐼𝑗) = 𝑓))
9796rexbidv 3182 . . . . . . 7 (𝑒 = 𝑓 → (∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑒 ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓))
9895, 97elab 3482 . . . . . 6 (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑒} ↔ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑓)
99 eqeq1 2756 . . . . . . 7 (𝑒 = 𝑓 → (𝑒 = {𝑁} ↔ 𝑓 = {𝑁}))
100 ushgredgedgloop.b . . . . . . 7 𝐵 = {𝑒𝐸𝑒 = {𝑁}}
10199, 100elrab2 3499 . . . . . 6 (𝑓𝐵 ↔ (𝑓𝐸𝑓 = {𝑁}))
10294, 98, 1013bitr4g 303 . . . . 5 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝑓 ∈ {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑒} ↔ 𝑓𝐵))
103102eqrdv 2750 . . . 4 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → {𝑒 ∣ ∃𝑗 ∈ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}} (𝐼𝑗) = 𝑒} = 𝐵)
10428, 103eqtr2d 2787 . . 3 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐵 = (𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}))
10519, 10, 104f1oeq123d 6286 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → (𝐹:𝐴1-1-onto𝐵 ↔ (𝐼 ↾ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}):{𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}–1-1-onto→(𝐼 “ {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}})))
1067, 105mpbird 247 1 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  {cab 2738  wrex 3043  {crab 3046  cdif 3704  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313  cmpt 4873  dom cdm 5258  ran crn 5259  cres 5260  cima 5261  Fun wfun 6035   Fn wfn 6036  wf 6037  1-1wf1 6038  1-1-ontowf1o 6040  cfv 6041  Vtxcvtx 26065  iEdgciedg 26066  Edgcedg 26130  UHGraphcuhgr 26142  USHGraphcushgr 26143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-edg 26131  df-uhgr 26144  df-ushgr 26145
This theorem is referenced by:  vtxdushgrfvedg  26588
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